Tilings · Year 10
In Tilings (Year 10) you practise 3 skills: The angle sum of a polygon · Angles of a regular polygon · Which polygons tile the plane. A tiling covers the plane with shapes, with no gaps and no overlaps. Round every vertex of a tiling the angles fill a full turn, 360°.
The diagonals from one vertex split a polygon into triangles, two fewer than it has sides. So the angles of a polygon add up to the number of sides minus 2, times 180°. In a regular polygon all the sides are equal and all the angles are equal. A regular polygon tiles the plane alone only when its angle goes into 360° a whole number of times: the equilateral triangle, the square and the regular hexagon. Every triangle and every quadrilateral tiles the plane. To find how many tiles are needed, put the lengths in the same unit and work out how many tiles fit along each side.
Tip: Before you work anything out, check every length is in the same unit: one metre is 100 cm.
What you learn
The angle sum of a polygon
For example: What is the sum of the interior angles of a polygon with 4 sides?
Answer and solution
The answer: 360 °
The diagonals from one vertex make 2 triangles, so the sum is (4 − 2) × 180° = 360°
Angles of a regular polygon
For example: What is the size of each interior angle of a square?
Answer and solution
The answer: 90 °
The angle sum: (4 − 2) × 180° = 360° . Each angle: 360° ÷ 4 = 90°
Which polygons tile the plane
For example: Each angle of a square is 90°. Can the plane be tiled with copies of it alone, with no gaps and no overlaps?
- No
- Yes
Answer and solution
The answer: Yes
The quotient 360 ÷ 90 = 4 is a whole number, so it tiles.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).